Operator Input Forms—Wolfram Language Documentation (original) (raw)

TECH NOTE

Operator Input Forms

Characters that are not letters, letter‐like forms, or structural elements are treated by the Wolfram Language as operators. The Wolfram Language has built‐in rules for interpreting all operators. The functions to which these operators correspond may or may not, however, have built‐in evaluation or other rules. Cases in which built‐in meanings are by default defined are indicated by ⊲ in the tables below.

Operators that construct two‐dimensional boxes—all of which have names beginning with backslash—can only be used inside \(…\). The table below gives the interpretations of these operators within ∖!∖(…∖). "Input of Boxes" gives interpretations when no \! is included.

Objects used in the tables of operator input forms.

Operator Precedence

operator form	full form	grouping
forms representing numbers (see Numbers )	⊲
forms representing symbols (see Symbol Names and Contexts )	⊲
forms representing character strings (see Character Strings )	⊲
e11e12… e21e22… …	{{e11,e12,…},{e21,e22,…},…}	⊲
 e11e12 e21e22 …	Piecewise[{{e11,e12},{e21,e22},…}]	⊲
expr::string	MessageName[expr,"string"]	⊲
expr::string1::string2	MessageName[expr,"string1","string2"]	⊲
forms containing # (see additional input forms)	⊲
forms containing % (see additional input forms)	⊲
forms containing _ (see additional input forms)	⊲
<<filename	Get["filename"]	⊲
	Overscript[expr1,expr2]
expr1\&expr2	Overscript[expr1,expr2]	e\&(e\&e)
	Underscript[expr1,expr2]
expr1\+expr2	Underscript[expr1,expr2]	e\+(e\+e)
	Underoverscript[expr1,expr2,expr3]
expr1\+expr2\%expr3	Underoverscript[expr1,expr2,expr3]
expr1\&expr2\%expr3	Underoverscript[expr1,expr3,expr2]
expr1expr2	Subscript[expr1,expr2]	e(ee)
expr1\_expr2	Subscript[expr1,expr2]	e\_(e\_e)
expr1\_expr2\%expr3	Power[Subscript[expr1,expr2],expr3]	⊲
\!boxes	(interpreted version of boxes )
expr1?expr2	PatternTest[expr1,expr2]		⊲
expr1[expr2,…]	expr1[expr2,…]	(e[e])[e]	⊲
expr1[[expr2,…]]	Part[expr1,expr2,…]	(e[[e]])[[e]]	⊲
expr1〚expr2,…〛	Part[expr1,expr2,…]	(e〚e〛)〚e〛	⊲
expr1〚expr2〛	Part[expr1,expr2,…]	(e〚e〛)〚e〛	⊲
expr1::[expr2,…]	TypeSpecifier[expr1][expr2,…]	(e::[e])::[e]	⊲
\*expr	(boxes constructed from expr )
expr++	Increment[expr]	⊲
expr--	Decrement[expr]	⊲
++expr	PreIncrement[expr]	⊲
--expr	PreDecrement[expr]	⊲
expr1@*expr2	Composition[expr1,expr2]	e@*e@*e	⊲
expr1/*expr2	RightComposition[expr1,expr2]	e/*e/*e	⊲
expr1expr2	Application[expr1,expr2]	(ee)e
expr1@expr2	expr1[expr2]	e@(e@e)	⊲
expr1 expr2	(invisible application, input as expr1 Esc@Esc expr2)	⊲
expr1[expr2]
expr1~expr2~expr3	expr2[expr1,expr3]	(e~e~e)~e~e	⊲
expr1/@expr2	Map[expr1,expr2]	e/@(e/@e)	⊲
expr1//@expr2	MapAll[expr1,expr2]	e//@(e//@e)	⊲
expr1@@expr2	Apply[expr1,expr2]	e@@(e@@e)	⊲
expr1@@@expr2	MapApply[expr1,expr2]	e@@@(e@@@e)	⊲
expr!	Factorial[expr]	⊲
expr!!	Factorial2[expr]	⊲
expr	Conjugate[expr]	⊲
expr	Transpose[expr]	⊲
expr	ConjugateTranspose[expr]	⊲
expr	ConjugateTranspose[expr]	⊲
expr'	Derivative[1][expr]	⊲
expr''…' (n times)	Derivative[n][expr]	⊲
expr1<>expr2<>expr3	StringJoin[expr1,expr2,expr3]	e<>e<>e	⊲
expr1^expr2	Power[expr1,expr2]	e^(e^e)	⊲
expr1expr2	Power[expr1,expr2]	e(ee)	⊲
	Power[Subscript[expr1,expr2],expr3]	⊲
expr1\^expr2\%expr3	Power[Subscript[expr1,expr3],expr2]	⊲
vertical arrow and vector operators
	Sqrt[expr]		⊲
\@ expr	Sqrt[expr]	\@(\@ e)	⊲
\@ expr\%n	Power[expr,1/n]	⊲
 expr	DifferentialD[expr]	( e)	⊲
∂expr1expr2	D[expr2,expr1]	∂e(∂ee)	⊲
∇ expr	Del[expr]	∇(∇e)
expr1expr2	DiscreteShift[expr2,expr1]	e(ee)	⊲
expr1expr2	DiscreteRatio[expr2,expr1]	e(ee)	⊲
expr1expr2	DifferenceDelta[expr2,expr1]	e(ee)	⊲
 expr	Square[expr]	( e)
expr1∘ expr2∘ expr3	SmallCircle[expr1,expr2,expr3]	e∘ e∘ e
expr1⊙ expr2⊙ expr3	CircleDot[expr1,expr2,expr3]	e ⊙ e ⊙ e
expr1**expr2**expr3	NonCommutativeMultiply[expr1,expr2,expr3]	e**e**e
expr1expr2expr3	Cross[expr1,expr2,expr3]	eee	⊲
expr1.expr2.expr3	Dot[expr1,expr2,expr3]	e.e.e	⊲
-expr	Times[-1,expr]	⊲
+expr	expr	⊲
±expr	PlusMinus[expr]
∓expr	MinusPlus[expr]
expr1/expr2	expr1(expr2)^-1	(e/e)/e	⊲
expr1÷expr2	Divide[expr1,expr2]	(e÷e)÷e	⊲
expr1\/expr2	Divide[expr1,expr2]	(e\/e)\/e	⊲
expr1∖expr2∖expr3	Backslash[expr1,expr2,expr3]	e∖e∖e
expr1⋄expr2⋄expr3	Diamond[expr1,expr2,expr3]	e⋄e⋄e
expr1⋀expr2⋀expr3	Wedge[expr1,expr2,expr3]	e⋀e⋀e
expr1⋁expr2⋁expr3	Vee[expr1,expr2,expr3]	e⋁e⋁e
expr1⊗expr2⊗expr3	CircleTimes[expr1,expr2,expr3]	e⊗e⊗e
expr1·expr2·expr3	CenterDot[expr1,expr2,expr3]	e·e·e
expr1 expr2 expr3	Times[expr1,expr2,expr3]	e e e	⊲
expr1*expr2*expr3	Times[expr1,expr2,expr3]	e*e*e	⊲
expr1×expr2×expr3	Times[expr1,expr2,expr3]	e×e×e	⊲
expr1⋆expr2⋆expr3	Star[expr1,expr2,expr3]	e⋆e⋆e
e4	Product[e4,{e1,e2,e3}]	∏(∏ e)	⊲
expr1≀expr2≀expr3	VerticalTilde[expr1,expr2,expr3]	e≀e≀e
expr1∐expr2∐expr3	Coproduct[expr1,expr2,expr3]	e∐e∐e
expr1⌢expr2⌢expr3	Cap[expr1,expr2,expr3]	e⌢e⌢e
expr1⌣expr2⌣expr3	Cup[expr1,expr2,expr3]	e⌣e⌣e
expr1⊕ expr2⊕ expr3	CirclePlus[expr1,expr2,expr3]	e⊕e⊕e
expr1⊖ expr2	CircleMinus[expr1,expr2]	(e ⊖ e)⊖ e
∫ expr1expr2	Integrate[expr1,expr2]	∫ (∫ e e) e	⊲
e3e4	Integrate[e3,{e4,e1,e2}]	∫ (∫ e e) e	⊲
∫e1∈e2e3	Integrate[e3,e1∈e2]	∫ (∫ e)	⊲
other integration operators
e4	Sum[e4,{e1,e2,e3}]	∑(∑ e)	⊲
e3	Limit[e3,e1e2]	(e)	⊲
e3	MaxLimit[e3,e1e2]	(e)	⊲
e3	MinLimit[e3,e1e2]	(e)	⊲
expr1+expr2+expr3	Plus[expr1,expr2,expr3]	e+e+e	⊲
expr1-expr2	expr1+(-1expr2)	e-e-e	⊲
expr1±expr2	PlusMinus[expr1,expr2]	(e±e)±e
expr1∓expr2	MinusPlus[expr1,expr2]	(e∓e)∓e
expr1⋂expr2	Intersection[expr1,expr2]	e⋂e⋂e	⊲
other intersection operators
expr1⋃expr2	Union[expr1,expr2]	e⋃e⋃e	⊲
other union operators
i;;j;;k	Span[i,j,k]	e;;e;;e	⊲
expr1==expr2	Equal[expr1,expr2]	e==e==e	⊲
expr1==expr2	Equal[expr1,expr2]	e==e==e	⊲
expr1expr2	Equal[expr1,expr2]	eee	⊲
expr1!= expr2	Unequal[expr1,expr2]	e!=e!=e	⊲
expr1!=expr2	Unequal[expr1,expr2]	e!=e!=e	⊲
other equality and similarity operators
expr1>expr2	Greater[expr1,expr2]	e>e>e	⊲
expr1>=expr2	GreaterEqual[expr1,expr2]	e>=e>=e	⊲
expr1≥expr2	GreaterEqual[expr1,expr2]	e≥e≥e	⊲
expr1⩾expr2	GreaterEqual[expr1,expr2]	e⩾e⩾e	⊲
expr1<expr2	Less[expr1,expr2]	e<e<e	⊲
expr1<=expr2	LessEqual[expr1,expr2]	e<=e<=e	⊲
expr1≤expr2	LessEqual[expr1,expr2]	e≤e≤e	⊲
expr1⩽expr2	LessEqual[expr1,expr2]	e⩽e⩽e	⊲
other ordering operators
expr1expr2	VerticalBar[expr1,expr2]	eee
expr1expr2	NotVerticalBar[expr1,expr2]	eee
expr1∥expr2	DoubleVerticalBar[expr1,expr2]	e∥e∥e
expr1∦expr2	NotDoubleVerticalBar[expr1,expr2]	e∦e∦e
horizontal arrow and vector operators
diagonal arrow operators
expr1===expr2	SameQ[expr1,expr2]	e===e===e	⊲
expr1=!=expr2	UnsameQ[expr1,expr2]	e=!=e=!=e	⊲
expr1∈expr2	Element[expr1,expr2]	e∈e∈e	⊲
expr1∉expr2	NotElement[expr1,expr2]	e∉e∉e	⊲
expr1⊂expr2	Subset[expr1,expr2]	e⊂e⊂e
expr1⊃expr2	Superset[expr1,expr2]	e⊃e⊃e
other set relation operators
∀expr1expr2	ForAll[expr1,expr2]	∀e(∀ee)	⊲
∃expr1expr2	Exists[expr1,expr2]	∃e(∃ee)	⊲
∄expr1expr2	NotExists[expr1,expr2]	∄e(∄ee)
!expr	Not[expr]	!(!e)	⊲
¬expr	Not[expr]	¬(¬e)	⊲
expr1&&expr2&&expr3	And[expr1,expr2,expr3]	e&&e&&e	⊲
expr1∧expr2∧expr3	And[expr1,expr2,expr3]	e∧e∧e	⊲
expr1⊼expr2⊼expr3	Nand[expr1,expr2,expr3]	e⊼e⊼e	⊲
expr1⊻expr2⊻expr3	Xor[expr1,expr2,expr3]	e⊻e⊻e	⊲
expr1expr2expr3	Xnor[expr1,expr2,expr3]	eee	⊲
expr1|	expr2		expr3
expr1∨expr2∨expr3	Or[expr1,expr2,expr3]	e∨e∨e	⊲
expr1⊽expr2⊽expr3	Nor[expr1,expr2,expr3]	e⊽e⊽e	⊲
expr1⧦expr2⧦expr3	Equivalent[expr1,expr2,expr3]	e⧦e⧦e	⊲
expr1expr2	Implies[expr1,expr2]	e(ee)	⊲
expr1⥰expr2	Implies[expr1,expr2]	e⥰e⥰e	⊲
expr1⊢expr2	RightTee[expr1,expr2]	e⊢(e⊢e)
expr1⊨expr2	DoubleRightTee[expr1,expr2]	e⊨(e⊨e)
expr1⊣expr2	LeftTee[expr1,expr2]	(e⊣e)⊣e
expr1⫤expr2	DoubleLeftTee[expr1,expr2]	(e⫤e)⫤e
expr1⊥expr2	UpTee[expr1,expr2]	(e⊥e)⊥e
expr1⊤expr2	DownTee[expr1,expr2]	(e⊤e)⊤e
expr1∍expr2	SuchThat[expr1,expr2]	e∍(e∍e)
expr..	Repeated[expr]	⊲
expr...	RepeatedNull[expr]	⊲
expr1|expr2	Alternatives[expr1,expr2]	e|e	e
symb:expr	Pattern[symb,expr]	⊲
symb:patt:expr	Optional[Pattern[symb,patt],expr]	⊲
patt:expr	Optional[patt,expr]	⊲
expr1~~expr2~~expr3	StringExpression[expr1,expr2,expr3]	e~~e~~e	⊲
expr1/;expr2	Condition[expr1,expr2]	(e/;e)/;e	⊲
expr1<->expr2	TwoWayRule[expr1,expr2]	e<->(e<->e)	⊲
expr1expr2	TwoWayRule[expr1,expr2]	e(ee)	⊲
expr1expr2	Rule[expr1,expr2]	e(ee)	⊲
expr1expr2	Rule[expr1,expr2]	e(ee)	⊲
expr1:>expr2	RuleDelayed[expr1,expr2]	e:>(e:>e)	⊲
expr1 expr2	RuleDelayed[expr1,expr2]	e(ee)	⊲
expr1/.expr2	ReplaceAll[expr1,expr2]	(e/.e)/.e	⊲
expr1//.expr2	ReplaceRepeated[expr1,expr2]	(e//.e)//.e	⊲
expr1+=expr2	AddTo[expr1,expr2]	e+=(e+=e)	⊲
expr1-=expr2	SubtractFrom[expr1,expr2]	e-=(e-=e)	⊲
expr1*=expr2	TimesBy[expr1,expr2]	e*=(e*=e)	⊲
expr1/=expr2	DivideBy[expr1,expr2]	e/=(e/=e)	⊲
expr&	Function[expr]	⊲
expr1∶expr2	Colon[expr1,expr2]	e∶e∶e
expr1//=expr2	ApplyTo[expr1,expr2]	e//=(e//=e)	⊲
expr1//expr2	expr2[expr1]	(e//e)//e
expr1expr2	VerticalSeparator[expr1,expr2]	eee
expr1∴expr2	Therefore[expr1,expr2]	e∴(e∴e)
expr1∵expr2	Because[expr1,expr2]	(e∵e)∵e
expr1=expr2	Set[expr1,expr2]	e=(e=e)	⊲
expr1:=expr2	SetDelayed[expr1,expr2]	e:=(e:=e)	⊲
expr1^=expr2	UpSet[expr1,expr2]	e^=(e^=e)	⊲
expr1^:=expr2	UpSetDelayed[expr1,expr2]	e^:=(e^:=e)	⊲
symb/:expr1=expr2	TagSet[symb,expr1,expr2]	⊲
symb/:expr1:=expr2	TagSetDelayed[symb,expr1,expr2]	⊲
expr=.	Unset[expr]	⊲
symb/:expr=.	TagUnset[symb,expr]	⊲
expr1|->expr2	Function[expr1,expr2]	e(ee)	⊲
expr1expr2	Function[expr1,expr2]	e(ee)	⊲
expr>>filename	Put[expr,"filename"]	⊲
expr>>>filename	PutAppend[expr,"filename"]	⊲
expr1;expr2;expr3	CompoundExpression[expr1,expr2,expr3]	⊲
expr1;expr2;	CompoundExpression[expr1,expr2,Null]	⊲
expr1\`expr2	FormBox[expr2,expr1]	e\`(e\`e)	⊲

Operator input forms, in order of decreasing precedence. Operators of equal precedence are grouped together.

Additional input forms, in order of decreasing precedence.

Special Characters

Special characters that appear in operators usually have names that correspond to the names of the functions they represent. Thus the character has the name ⊕ and yields the function CirclePlus. Exceptions are ⩾, ⩽ and ⥰.

The delimiters in matchfix operators have names \[LeftName] and \[RightName].

"Listing of Named Characters" gives a complete listing of special characters that appear in operators.

Keyboard and special characters with the same interpretations.

| keyboard characterspecial character : :∶ ∶ ~ ~∼ ∼ ^ ^⋀ ⋀ ^ ^∧ ∧ * *⋆ ⋆ \ ∖∖ ∖ | keyboard characterspecial character . .· · | |   | |   | |   - -– – ...… … | | -------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- | ----------------------------------------------------------------------------------------------------------------------------------------------------------------------------- | ------------------------------------------------------------------------------------------------ | ------------------------------------------------------------------------------------------------------ | ------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- |

Some keyboard and special characters with different interpretations.

Precedence and the Ordering of Input Forms

Grouping of Input Forms

The third columns in the tables show how multiple occurrences of a single input form, or of several input forms with the same precedence, are grouped. For example, a/b/c is grouped as (a/b)/c ("left associative"), while a^b^c is grouped as a^(b^c) ("right associative"). No grouping is needed in an expression like a+b+c, since Plus is fully associative, as represented by the attribute Flat.

Precedence of Integration Operators

∮, ∲ and ∯ work the same as ∫.

See "Two-Dimensional Input Forms" for two‐dimensional input forms associated with integration operators.

Spaces and Multiplication

Spaces in the Wolfram Language denote multiplication, just as they do in standard mathematical notation. In addition, the Wolfram Language takes complete expressions that are adjacent, not necessarily separated by spaces, to be multiplied together.

Alternative forms for multiplication.

An expression like x!y could potentially mean either (x!)*y or x*(!y). The first interpretation is chosen because Factorial has higher precedence than Not.

Spaces within single input forms are ignored. Thus, for example, a + b is equivalent to a+b. You will often want to insert spaces around lower precedence operators to improve readability.

You can give a "coefficient" for a symbol by preceding it with any sequence of digits. When you use numbers in bases larger than 10, the digits can include letters. (In bases other than 10, there must be a space between the end of the coefficient and the beginning of the symbol name.)

Some cases to be careful about.

Spaces to Avoid

You should avoid inserting any spaces between the different characters in composite operators such as /., =., and >=. Although in some cases such spaces are allowed, they are liable to lead to confusion.

Another case where spaces must be avoided is between the characters of the pattern object x_. If you type x_, the Wolfram Language will interpret this as x*_, rather than the single named pattern object x_.

Similarly, you should not insert any spaces inside pattern objects like x_:value.

Spacing Characters

Spacing characters equivalent to an ordinary keyboard space.

Relational Operators

Relational operators can be mixed. An expression like a>b>=c is converted to Inequality[a,Greater,b,GreaterEqual,c], which effectively evaluates as (a>b)&&(b>=c). (The reason for the intermediate Inequality form is that it prevents objects from being evaluated twice when input like a>b>=c is processed.)

File Names

Any file name can be given in quotes after <<, >>, and >>>. File names can also be given without quotes if they contain only alphanumeric characters and the characters `, /, ., ∖, !, -, _, :, $, *, ~, and ?, together with matched pairs of square brackets enclosing any characters other than spaces, tabs, and newlines. Note that file names given without quotes can be followed only by spaces, tabs, or newlines, or by the characters ), ], or }, as well as semicolons and commas.